06vt Factoring Polynomials.notebook
September 20, 2017
Factoring Polynomials
Remainder Theorem: When a polynomial,
, the remainder is equal to
divided by
, is
.
Factor Theorem:
If the remainder, or
, is equal to zero, then
.
is a factor of the polynomial
For example, recall:
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Ex.1 Use the factor theorem to determine one factor of
then completely factor the function.
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1
06vt Factoring Polynomials.notebook
September 20, 2017
A rational number can be expressed as a fraction with
an integer numerator and denominator (but no division
by zero).
where
A rational root is a zero which is a rational number. For a
polynomial, roots can be expressed as factors:
, or, more commonly,
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The rational roots test allows us to limit our search for roots
using the leading term and the constant (last) term.
For a polynomial in the form
the possible rational roots are
all factors of constant term
all factors of leading coefficient
note: Some roots are irrational, and there is no guarantee
that the rational root test will be successful.
Therefore, not all polynomials will be factorable.
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06vt Factoring Polynomials.notebook
September 20, 2017
Sep 23­6:14 PM
For example,
and a leading coefficient of 3.
has a constant term ­8
factors of 8 are 1, 2, 4, 8
factors of 3 are 1, 3
possible rational roots are
As a list:
Using the factor theorem, we can test each one of these
until
. For this quadratic,
,
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3
06vt Factoring Polynomials.notebook
September 20, 2017
Ex.2 Determine all possible rational roots for
then show that
is a factor.
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Assigned Work:
p.176 #
[1 ­ 3][basics],
4bf, 5ac, 6abc, 7df, 8, 9, 10, 13, 17
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